When coherent (laser) light passes through or reflects from an object, its amplitude and phase are altered as it continues to propagate. Phase perturbations contain important information about an object. For example, transparent biological cells are invisible in a focused microscope, but impart distinct phase changes. Data about these phase changes can find useful applications in morphology analysis of cell, or cell-cycle analysis without labelling. Such applications are relevant for biology and biomedical engineering, e.g., for detecting cancer cells. Moreover, there are various applications in materials science.
It is noteworthy that only the intensity of light can be measured directly, since phase oscillates far too quickly, and therefore, the phase needs to be reconstructed computationally. This “phase problem” of optical imaging has been around for decades, but only recently has been posed as an inference problem [1, 2]. This breakthrough, in principle, allows for much more experimental noise in the images, and as a consequence, enables applications of phase imaging where there is little light and much background noise (e.g., underwater imaging).
Traditional methods for phase recovery include phase contrast microscopy, differential interference contrast microscopy, and digital holography [3,4,5]. All of the three methods use a reference wavefront to obtain the amplitude and phase information. The phase image recovered by phase contrast microscopy is a function of the optical path length magnitude of the object. Differential interference contrast microscopy obtains the gradients of the optical length, but it can only be used when the object has a similar refractive index to the surroundings. Digital holography uses an interferometer setup to record the interference between a reference beam and a beam which has interacted with an object to be imaged. A computer is used to calculate the object image with a numerical reconstruction algorithm. So digital holography has the advantage of giving quantifiable information about optical distance, while phase contrast microscopy and differential interference contrast microscopy just provide a distorting of the bright field image with phase shift information. However, the experimental setup for digital holography is usually complicated, and has high requirements on the wave path [6,7]. For instance, the reference beam and target beam need to be accurately aligned.
An alternative approach is based on exploiting the physics of wavefront propagation. Consider the experimental arrangement shown in FIG. 1, which is taken from [1]. A laser 1 generates a beam 2 of light, which propagates in a direction z and passes through a collimator 3, and a lens 4. An object to be imaged can be placed in the object plane 5, such that the light beam passes through it. The light then passes through a further pair of lenses lens 6, 7 forming a “4f system”, and the light then passes into a camera 8 which collects an intensity image of the light reaching it in the propagation direction z. The 4f system has the effect of enlarging the beam (which is important if the object which was imaged is small, such as a cell) and of making it clear where the position of the object plane is in relation to the focal plane of the lenses 6, 7 (this information is important, so that it is possible to propagate the complex plane back into the object plane) The camera 8 or the object 5 can be moved parallel to the light propagation direction, and collects a plurality of two-dimensional intensity images 9 at respective locations spaced apart in this direction. Each intensity image shows the light intensity at respective points in a two-dimensional plane (x,y) perpendicular to the light propagation direction z. Note that the distance by which the camera or object 5 is moved is very small (of the order of 10-100 micrometers) so there is no collision with the lenses. The focal plane is determined by the focal length of the lenses 6, 7 in the 4f system. The complex field at the focal plane is an enlarged or reduced version of the complex field at the object plane 5, so recovering the complex field at the focal plane is the same as recovering the complex field at the plane 5. At the focal plane, the intensity image 9a contains no information about the phase.
The object to be imaged, which is placed at the object plane 5, modifies the light passing through it, producing, at each point in the object plane 5, a corresponding amplitude contrast and phase difference. In example, the phase difference produced by the object at each point in the 2-D object plane is as shown as 10a in FIG. 1, where the level of brightness indicates a corresponding phase difference. The amplitude contrast produced by the object in the object plane is shown as 10b (a moustache and hat). The amplitude contrast diffracts symmetrically through the focal point, while the phase defocuses in an anti-symmetric fashion. The phase and amplitude are estimated from the intensity images 9 captured by the camera 4 by a computational numerical algorithm.
One method for doing this is the Gerchberg-Saxton (GS) method [9.10], which treats the problem as convex optimization and iterates back and forth between two domains (an in-focus image and a Fourier domain image) to reduce error at each iteration. It is strongly sensitive to the noise in the latter image. An alternative method is a direct method [11, 12, 13] which exploits the Transport of Intensity Equation (TIE); it is based on first- and higher-order derivatives, and it is not robust to noise. Thus, although the GS and direct methods are computationally efficient, they are both very sensitive to noise.
A few statistical approaches have been proposed as well; an approximation to the maximum likelihood estimator is derived in [2, 14]. However, it easily gets stuck in local maxima, and sometimes leads to poor results. In [1] an augmented complex extended Kalman filter (ACEKF) was used to solve for phase with significant noise corruption.
However, the memory requirements are of order N2 where N is the number of pixels in each intensity image, which is unfeasible for practical image sizes of multiple megapixels, and the long computation times are impractical for real-time applications, such as in biology, biomedical engineering and beyond. In [8] a diagonalized complex extended Kalman filter (diagonalized CEKF) was proposed to alleviate those issues, without jeopardizing the reconstruction accuracy. The diagonalized CEKF is iterative: it needs to cycle through the set of intensity images repeatedly, yielding more accurate phase reconstruction after each cycle. On the other hand, the computational complexity increases with each cycle.